$\underset{x\to 0}{lim}f\left(x\,m x\right)$

$\=\underset{x\to 0}{lim}\frac{x^2\+ m x}{\sqrt{x^2\+m^2{x}^2}}$

$\=\underset{x\to 0}{lim}\frac{x\+m}{\sqrt{1\+m^2}}$

$\=\frac{m}{\sqrt{1\+m^2}}$

Define the function $f\left(x\,y\right)$ 

$f\left(x\,y\right)\=\frac{x^2\+y}{\sqrt{x^2\+y^2}}$$\stackrel{\text{assign as function}}{\to }$$f$

Simplify $f\left(x\,m x\right)$ under the assumption that $x\>0$ 

$f\left(x\,m x\right)$

$\stackrel{\text{assuming positive}}{\to }$

Evaluate $\underset{x\to 0}{lim}f\left(x\,m x\right)$

$\underset{x\to 0}{lim}$ = $\frac{m}{\sqrt{m^2\+1}}$

$f≔\left(x\,y_{}\right)\to \frac{x^2\+y}{\sqrt{x^2\+y^2}}\:$

$\mathrm{simplify}\left(f\left(x\,m x\right)\right) assuming x\>0$

$\mathrm{limit}\left(\,x\=0\right)$ = $\frac{m}{\sqrt{m^2\+1}}$